Optimal. Leaf size=55 \[ \frac{3}{8} \cosh (a) \text{Chi}\left (b x^2\right )+\frac{1}{8} \cosh (3 a) \text{Chi}\left (3 b x^2\right )+\frac{3}{8} \sinh (a) \text{Shi}\left (b x^2\right )+\frac{1}{8} \sinh (3 a) \text{Shi}\left (3 b x^2\right ) \]
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Rubi [A] time = 0.0942874, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5341, 5319, 5317, 5316} \[ \frac{3}{8} \cosh (a) \text{Chi}\left (b x^2\right )+\frac{1}{8} \cosh (3 a) \text{Chi}\left (3 b x^2\right )+\frac{3}{8} \sinh (a) \text{Shi}\left (b x^2\right )+\frac{1}{8} \sinh (3 a) \text{Shi}\left (3 b x^2\right ) \]
Antiderivative was successfully verified.
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Rule 5341
Rule 5319
Rule 5317
Rule 5316
Rubi steps
\begin{align*} \int \frac{\cosh ^3\left (a+b x^2\right )}{x} \, dx &=\int \left (\frac{3 \cosh \left (a+b x^2\right )}{4 x}+\frac{\cosh \left (3 a+3 b x^2\right )}{4 x}\right ) \, dx\\ &=\frac{1}{4} \int \frac{\cosh \left (3 a+3 b x^2\right )}{x} \, dx+\frac{3}{4} \int \frac{\cosh \left (a+b x^2\right )}{x} \, dx\\ &=\frac{1}{4} (3 \cosh (a)) \int \frac{\cosh \left (b x^2\right )}{x} \, dx+\frac{1}{4} \cosh (3 a) \int \frac{\cosh \left (3 b x^2\right )}{x} \, dx+\frac{1}{4} (3 \sinh (a)) \int \frac{\sinh \left (b x^2\right )}{x} \, dx+\frac{1}{4} \sinh (3 a) \int \frac{\sinh \left (3 b x^2\right )}{x} \, dx\\ &=\frac{3}{8} \cosh (a) \text{Chi}\left (b x^2\right )+\frac{1}{8} \cosh (3 a) \text{Chi}\left (3 b x^2\right )+\frac{3}{8} \sinh (a) \text{Shi}\left (b x^2\right )+\frac{1}{8} \sinh (3 a) \text{Shi}\left (3 b x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0333854, size = 49, normalized size = 0.89 \[ \frac{1}{8} \left (3 \cosh (a) \text{Chi}\left (b x^2\right )+\cosh (3 a) \text{Chi}\left (3 b x^2\right )+3 \sinh (a) \text{Shi}\left (b x^2\right )+\sinh (3 a) \text{Shi}\left (3 b x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 55, normalized size = 1. \begin{align*} -{\frac{{{\rm e}^{-3\,a}}{\it Ei} \left ( 1,3\,b{x}^{2} \right ) }{16}}-{\frac{3\,{{\rm e}^{-a}}{\it Ei} \left ( 1,b{x}^{2} \right ) }{16}}-{\frac{{{\rm e}^{3\,a}}{\it Ei} \left ( 1,-3\,b{x}^{2} \right ) }{16}}-{\frac{3\,{{\rm e}^{a}}{\it Ei} \left ( 1,-b{x}^{2} \right ) }{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.31971, size = 68, normalized size = 1.24 \begin{align*} \frac{1}{16} \,{\rm Ei}\left (3 \, b x^{2}\right ) e^{\left (3 \, a\right )} + \frac{3}{16} \,{\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + \frac{1}{16} \,{\rm Ei}\left (-3 \, b x^{2}\right ) e^{\left (-3 \, a\right )} + \frac{3}{16} \,{\rm Ei}\left (b x^{2}\right ) e^{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66289, size = 231, normalized size = 4.2 \begin{align*} \frac{1}{16} \,{\left ({\rm Ei}\left (3 \, b x^{2}\right ) +{\rm Ei}\left (-3 \, b x^{2}\right )\right )} \cosh \left (3 \, a\right ) + \frac{3}{16} \,{\left ({\rm Ei}\left (b x^{2}\right ) +{\rm Ei}\left (-b x^{2}\right )\right )} \cosh \left (a\right ) + \frac{1}{16} \,{\left ({\rm Ei}\left (3 \, b x^{2}\right ) -{\rm Ei}\left (-3 \, b x^{2}\right )\right )} \sinh \left (3 \, a\right ) + \frac{3}{16} \,{\left ({\rm Ei}\left (b x^{2}\right ) -{\rm Ei}\left (-b x^{2}\right )\right )} \sinh \left (a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh ^{3}{\left (a + b x^{2} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27431, size = 68, normalized size = 1.24 \begin{align*} \frac{1}{16} \,{\rm Ei}\left (3 \, b x^{2}\right ) e^{\left (3 \, a\right )} + \frac{3}{16} \,{\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + \frac{1}{16} \,{\rm Ei}\left (-3 \, b x^{2}\right ) e^{\left (-3 \, a\right )} + \frac{3}{16} \,{\rm Ei}\left (b x^{2}\right ) e^{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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